[Paper Review] Network Flow and Copper Plate Relaxations for AC Transmission Systems

Carleton Coffrin, Hassan Hijazi+1 more|arXiv (Cornell University)|Jun 17, 2015

Optimal Power Flow Distribution5 references4 citations

TL;DR

This paper introduces two novel linear relaxations—Network Flow (NF) and Copper Plate (CP)—for AC transmission system optimal power flow, derived from physical power flow principles. The NF relaxation is theoretically proven to be a relaxation of the second-order cone (SOC) relaxation, while CP further relaxes NF, offering a trade-off between computational efficiency and solution quality, with numerical results showing CP and NF gaps of 2.99% and 5.90% respectively on a 3-bus system, significantly better than a prior linear method (87.2% gap).

ABSTRACT

Nonlinear convex relaxations of the power flow equations and, in particular, the Semi-Definite Programming (SDP), Convex Quadratic (QC), and Second-Order Cone (SOC) relaxations, have attracted significant interest in recent years. Thus far, little attention has been given to simpler linear relaxations of the power flow equations, which may bring significant performance gains at the cost of model accuracy. To fill the gap, this paper develops two intuitive linear relaxations of the power flow equations, one based on classic network flow models (NF) and another inspired by copper plate approximations (CP). Theoretical results show that the proposed NF model is a relaxation of the established nonlinear SOC model and the CP model is a relaxation of the NF model. Consequently, considering the linear NF and CP relaxations alongside the established nonlinear relaxations (SDP, QC, SOC) provides a rich variety of tradeoffs between the relaxation accuracy and performance.

Motivation & Objective

• To address the lack of attention on linear relaxations in AC power flow optimization, despite their potential for improved performance and scalability.
• To develop two intuitive, physically motivated linear relaxations—Network Flow (NF) and Copper Plate (CP)—for AC transmission systems.
• To establish theoretical relationships showing that NF is a relaxation of the established SOC relaxation and CP is a relaxation of NF.
• To evaluate the trade-offs between relaxation strength (optimality gap) and computational efficiency across multiple relaxation types.
• To demonstrate through numerical experiments that the proposed relaxations outperform existing linear methods in solution quality while maintaining linear complexity.

Proposed method

• Proposes a Network Flow (NF) relaxation based on linearized power flow equations derived from Kirchhoff’s Current Law and Ohm’s Law, using voltage magnitude squared (W) and current magnitude squared (L) as variables.
• Develops a Copper Plate (CP) relaxation inspired by the copper plate approximation, modeling active and reactive power flows as linear functions of voltage magnitudes and line parameters.
• Proves that the NF relaxation is a valid relaxation of the second-order cone (SOC) relaxation by showing that all feasible solutions to SOC are also feasible in NF.
• Demonstrates that the CP relaxation is a relaxation of the NF model by introducing additional linear constraints that further loosen the feasible region.
• Uses a 3-bus test system with detailed network parameters (including shunts, charging, transformers, and limits) to compare relaxations numerically.
• Employs a global optimization solver (Couenne) to compute the true optimal AC power flow solution and computes optimality gaps as a performance metric.

Experimental results

Research questions

• RQ1 Can simple linear relaxations of AC power flow provide a viable alternative to nonlinear convex relaxations in terms of computational efficiency and scalability?
• RQ2 How do the proposed NF and CP relaxations compare in solution quality (optimality gap) to established nonlinear relaxations like SOC and SDP?
• RQ3 What is the theoretical relationship between the proposed NF and CP relaxations and existing nonlinear relaxations such as SOC?
• RQ4 How do the proposed relaxations compare to prior linear relaxations (e.g., TH model) in terms of solution quality and feasibility?
• RQ5 What is the impact of system constraints (e.g., phase angle differences) on the performance of linear relaxations in AC power flow problems?

Key findings

• The NF relaxation is a valid relaxation of the established SOC relaxation, ensuring that all feasible solutions to SOC are also feasible in NF.
• The CP relaxation is a valid relaxation of the NF model, further expanding the hierarchy of relaxations from nonlinear to linear approximations.
• On the 3-bus test case with standard constraints, the NF and CP relaxations achieve optimality gaps of 2.99% and 5.90%, respectively, significantly outperforming the prior linear method (TH) with an 87.2% gap.
• When phase angle difference limits are tightened to 18°, the optimality gaps for NF and CP increase to 5.90%, showing sensitivity to system constraints.
• The SOC relaxation remains the tightest among all relaxations tested, with a 1.32% optimality gap in the base case, confirming its superior accuracy.
• The proposed relaxations offer a compelling trade-off between computational speed and solution quality, making them suitable for large-scale or real-time applications.